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On Moran's Property of the Poisson Distribution

Panaretos, John (1983): On Moran's Property of the Poisson Distribution. Published in: Biometrical Journal , Vol. Vol.25, No. 1 (1983): pp. 69-76.

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Two interesting results encountered in the literature concerning the Poisson and the negative binomial distributions are due to MORAN (1952) and PATIL & SESHADRI (1964), respectively.

MORAN's result provided a fundamental property of the Poisson distribution. Roughly speaking, he has shown that if Y, Z are independent, non-negative, integer-valued random variables with X=Y | Z then, under some mild restrictions, the conditional distribution of Y | X is binomial if and only if Y, Z are Poisson random variables.

Motivated by MORAN's result PATIL & SESHADRI obtained a general characterization. A special case of this characterization suggests that, with conditions similar to those imposed by MORAN, Y | X is negative hypergeometric if and only if Y, Z are negative binomials.

In this paper we examine the results of MORAN and PATIL & SESHADRI in the case where the conditional distribution of Y | X is truncated at an arbitrary point k-1 (k=1, 2, …). In fact we attempt to answer the question as to whether MORAN's property of the Poisson distribution, and subsequently PATIL & SESHADRI's property of the negative binomial distribution, can be extended, in one form or another, to the case where Y | X is binomial truncated at k-1 and negative hypergeometric truncated at k-1 respectively

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